You are a bug sitting in one corner of a cubic room. You wish to walk (not fly) to the extreme opposite corner – the corner furthest from you. Describe the shortest path that you can walk.
Refer to the diagram.
The problem phrased differently is that we have to get from point A to point B only moving along the walls.
The shortest route is shown it is A-H-B where H is the mid point of D-E.
The length of this route can easily be calculated, assume the cube has sides of length 1 unit (it doesn’t matter what these units are, meters, feet, what ever) The distance A-H is the hypotenuse of a triangle 1 x ½ a quick bit of pythag tells us that A-H equals sqrt(5/4). Similarly H-B has the same length hence the total length is 2 x sqrt(5/4) this is actually equal to the square root of 5
A-H-B = sqrt(5) = 2.236
Some people think the shortest root is A-C-B or A-E-B or A-F-B etc. (they are all the same) this has a length of 1 + sqrt(2) ie. about 2.414.